Let K be a purely inseparable extension of a field k of characteristic p ≠ 0. Suppose that is finite. We recall that K/k is lq-modular if K is modular over a finite extension of k. Moreover, there exists a smallest extension m/k (resp. M/K) such that K/m (resp. M/k) is lq-modular. Our main result states the existence of a greatest lq-modular and relatively perfect subextension of K/k. Other results can be summarized in the following: 1. The product of lq-modular extensions over k is lq-modular over k. 2. If we augment the ground field of an lq-modular extension, the lq-modularity is preserved. Generally, for all intermediate fields K₁ and K₂ of K/k such that K₁/k is lq-modular over k, K₁(K₂)/K₂ is lq-modular. By successive application of the theorem on lq-modular closure (our main result), we deduce that the smallest extension m/k of K/k such that K/m is lq-modular is non-trivial (i.e. m ≠ K). More precisely if K/k is infinite, then K/m is infinite.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-cm122-2-13,
author = {Mustapha Chellali and El hassane Fliouet},
title = {Th\'eor\`eme de la cl\^oture lq-modulaire et applications},
journal = {Colloquium Mathematicae},
volume = {122},
year = {2011},
pages = {275-287},
zbl = {1239.12005},
language = {fra},
url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm122-2-13}
}
Mustapha Chellali; El hassane Fliouet. Théorème de la clôture lq-modulaire et applications. Colloquium Mathematicae, Tome 122 (2011) pp. 275-287. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-cm122-2-13/